If a card in a player's graveyard has in its mana cost, X is considered to be 0.

The last ability of Idol of Endurance doesn't change when you may cast those creature spells. Normally this means during your main phase while the stack is empty, but you can cast a creature spell with flash at other times.

If you cast a spell with Idol of Endurance's last ability, it becomes a new object. If it returns to exile, it's no longer exiled with Idol of Endurance, and won't be returned to your graveyard once Idol of Endurance leaves the battlefield.

If you cast a spell “without paying its mana cost,” you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, those must be paid to cast the spell.

If a spell has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

The exiled cards return to your graveyard immediately after Idol of Endurance leaves the battlefield. Any cards that had left exile before then (most likely because they were cast with Idol of Endurance's second ability) remain in their current zone.